Find the remainder when the polynomial $x^{1000}$ is divided by the polynomial $(x^2 + 1)(x + 1).$
Answer: Note that $(x^2 + 1)(x + 1)$ is a factor of $(x^2 + 1)(x + 1)(x - 1) = x^4 - 1.$  Since
\[x^{1000} - 1 = (x^4 - 1)(x^{996} + x^{992} + x^{988} + \dots + x^8 + x^4 + 1),\]the remainder when $x^{1000}$ is divided by $(x^2 + 1)(x + 1)$ is $\boxed{1}.$